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Worksheet: Resultant Couple

In this worksheet, we will practice finding the resultant of two or more couples.

Q1:

𝐴𝐵𝐶𝐷 is a square with sides of length 28 cm. Two forces of magnitude
117 N act along 𝐴𝐵 and 𝐶𝐷,
and two other forces of magnitude of 177 N act along 𝐴𝐷 and 𝐶𝐵 as shown in the figure.
Find the algebraic measure of the moment of the resultant couple.

Q2:

𝐴𝐵𝐶 is a lamina in the shape of an isosceles triangle,
where 𝐴𝐵=𝐴𝐶=13cm and 𝐵𝐶=10cm.
It rotates freely in a vertical plane about a fixed hinge at 𝐴.
A couple with moment of magnitude 67 N⋅cm acts on
the lamina to keep it in equilibrium with 𝐴𝐵 vertical.
Find the weight of the lamina given that it acts at the point of intersection of its medians.

A43.55 N

B8.38 N

C16.75 N

D21.77 N

Q3:

𝐴𝐵𝐶𝐷 is a square with a side length of 18 cm,
where forces of magnitudes 32 N,
𝐹 N, 32 N,
and 𝐹 N are acting along 𝐵𝐴, 𝐵𝐶, 𝐷𝐶, and 𝐷𝐴,
respectively. If the four forces are equivalent to a couple, where the magnitude of its moment is 90 N⋅cm in the direction of 𝐴𝐷𝐶𝐵, determine the magnitude of 𝐹.

Q4:

𝐴𝐵𝐶𝐷 is a rectangle, in which 𝐴𝐵=14cm,
and 𝐵𝐶=24cm. Given that 𝑋 and
𝑌 are the midpoints of 𝐴𝐵 and
𝐶𝐷, respectively, and forces of magnitudes 153 N, 199 N, 153 N, 199 N, 73 N, and 73 N are acting as shown in the
figure, determine the moment of the resultant couple rounded to two decimal places.

Q5:

is a right triangular lamina, with the right-angle at
. Length , length
, and its weight is 48 N. The lamina’s weight is acting at the point of intersection of its medians
(lines drawn from each vertex to the midpoint of the opposite side), and the lamina is
suspended from vertex , where its plane is vertical. Determine the
magnitude of the moment of the couple that would hold in the
vertical position.

A648 N⋅cm

B864 N⋅cm

C

1 296

N⋅cm

D432 N⋅cm

Q6:

𝐴𝐵𝐶𝐷 is a rectangle, where 𝐴𝐵=8cm and 𝐵𝐶=6cm.
If forces of magnitudes 35, 9, 35, and
9 newtons are acting along 𝐵𝐴, 𝐵𝐶,
𝐷𝐶, and 𝐷𝐴, respectively, determine their moment 𝑀.
Also, for the system to be in equilibrium, another two forces, each of magnitude 𝐹, are required to act at 𝐴
and 𝐶, where their lines of action are perpendicular to 𝐴𝐶. Determine the value of 𝐹
and the direction of rotation of the formed couple.

A𝑀=−138⋅Ncm, 𝐹=13.8N, acting clockwise

B𝑀=−282⋅Ncm, 𝐹=28.2N, acting counterclockwise

C𝑀=282⋅Ncm, 𝐹=28.2N, acting clockwise

D𝑀=−138⋅Ncm, 𝐹=13.8N, acting counterclockwise

Q7:

The forces , , and are acting at the
points , , and
respectively, where
,
, and
. Given that another force
of magnitude 50 N is acting at , where the four forces are
equivalent to a couple, determine the magnitude of the moment of the couple.

Q8:

The forces , , and are acting at the
points , , and
respectively, where
,
, and
. Given that another force
of magnitude 60 N is acting at , where the four forces are
equivalent to a couple, determine the magnitude of the moment of the couple.

Q9:

𝐴𝐵𝐶𝐷 is a parallelogram, in which 𝐴𝐵=10cm,
𝐵𝐶=10cm, and 𝑚∠𝐴𝐵𝐶=150∘. Forces of magnitudes 3, 6,
3, and 6 newtons are acting along 𝐴𝐵,
𝐶𝐵, 𝐶𝐷, and 𝐴𝐷, respectively. Find the moment of the resulting couple.

A45 N⋅cm

B30 N⋅cm

C90 N⋅cm

D15 N⋅cm

Q10:

Given that the forces in the figure have magnitudes of 56, 105, and 242 newtons, find the algebraic measure of the resultant couple.

Q11:

𝐴𝐵𝐶 is a triangle, where 𝐴𝐵=20cm,
𝐵𝐶=25cm,
and 𝐶𝐴=15cm,
and forces of magnitudes 120, 150, and 90
newtons are acting along 𝐴𝐵, 𝐵𝐶, and 𝐶𝐴 respectively,
where the system is equivalent to a couple. Determine the magnitudes of the two parallel forces that would make the system in equilibrium when acting at 𝐵 and 𝐶 perpendicularly to 𝐵𝐶.

A36 N, 36 N

B144 N, 144 N

C72 N, 72 N

Q12:

A lamina in the shape of a square has a side length of 20 cm
and a weight of 15 N, which is acting at
the point of intersection of its diagonals. The lamina is suspended from a horizontal pin
near its vertex 𝐴 such that its plane is vertical, and a couple is acting
on the lamina that makes it in equilibrium in a position where
𝐴𝐶 is inclined to the vertical at an angle of
30∘.
Determine the magnitude of the moment of the
couple.

A150 N⋅cm

B75√3 N⋅cm

C75√6 N⋅cm

D75√2 N⋅cm

Q13:

is a lamina in the form of an equilateral triangle weighing 27 N that is acting at the point of intersection of the medians. The lamina is suspended by a thin horizontal pin passing through a small hole
near the vertex , thus the plane is vertical as shown in the figure. Also, a couple of moment
50 N⋅cm is acting on the lamina perpendicularly to its plane.
Given that the height of the lamina is 9 cm,
find the size of the acute angle that makes with the horizontal when the lamina is in equilibrium rounded to the nearest minute.

A

B

C

D

Q14:

𝐴𝐵𝐶𝐷 is a rectangle, where 𝐴𝐵=24cm, and 𝐵𝐶=7cm.
Two forces, each of magnitude 43 N, are acting along 𝐵𝐴 and 𝐷𝐶, respectively.
Determine the magnitude of each of the two forces acting at points 𝐵 and 𝐷 and perpendicular to 𝐵𝐷 that would make the whole system in equilibrium.

A41.28 N, 41.28 N

B24.08 N, 24.08 N

C82.56 N, 82.56 N

D12.04 N, 12.04 N

Q15:

𝐴𝐵𝐶𝐷𝐸𝑂 is a regular hexagon, where forces of magnitudes 12√3 N, 81 N,
12√3 N, and 81 N are acting along 𝐴𝐵,
𝐷𝐵, 𝐷𝐸, and 𝐴𝐸, respectively.
If the system is equivalent to a couple, determine the magnitude of each of the two forces acting along 𝐸𝐵 and
𝐴𝑂 for the system to be in equilibrium.

A78√3 N, 78√3 N

B15√3 N, 15√3 N

C39√3 N, 39√3 N

D30√3 N, 30√3 N

Q16:

A rod 𝐴𝐵 has length 5 cm and weight 6 N, which is acting at its midpoint. It can rotate freely in a vertical plane about a horizontal pin at 𝐶, where 𝐴𝐶=2cm. A force of magnitude 6 N is acting at 𝐴 vertically upwards. Determine the magnitude of the force 𝐹 that brings the rod to equilibrium when it acts at 𝐵 perpendicularly to 𝐴𝐵. Also find the reaction of the pin 𝑅 when the rod is inclined to the horizontal at an angle of 45∘ such that 𝐴 is higher than 𝐵.

A𝐹=3√22N, 𝑅=3√22N at
𝐶, perpendicular to 𝐴𝐵 downwards

B𝐹=5√2N, 𝑅=5√2N at
𝐶, perpendicular to 𝐴𝐵 upwards

C𝐹=15√22N, 𝑅=15√22N at
𝐶, perpendicular to 𝐴𝐵 upwards

D𝐹=5√22N, 𝑅=5√22N at
𝐶, perpendicular to 𝐴𝐵 downwards

Q17:

𝐴𝐵𝐶 is a uniform lamina in the form of an equilateral triangle with side length
35 cm.
Forces of 41 N, 28 N, and
61 N act perpendicularly on 𝐴𝐵, 𝐵𝐶, and
𝐶𝐴 respectively as shown in the figure.
Find the algebraic measure of the moment of the resultant couple.

Q18:

In a square𝐴𝐵𝐶𝐷, whose side length is 3𝐿,
𝐹∈𝐴𝐵, 𝐸∈𝐷𝐶 and 𝐹𝐵=𝐸𝐷=1√3𝐿.
Two forces, each of magnitude 143 newtons, are acting along 𝐴𝐷 and 𝐶𝐵.
Two other forces, each of magnitude 164 newtons, are acting along 𝐹𝐶 and 𝐸𝐴.
Given that the moment of the resultant couple equals 𝑐𝐿 N⋅cm, determine the value of 𝑐 to the nearest integer.

Q19:

A uniform rod 𝐴𝐵 having a length of 85 cm
and a weight of 10 N is moving in a
vertical plane about a fixed hinge at its end 𝐴. Given that a couple of
moment 130 N⋅cm is acting on the rod and
perpendicular to its plane, find the inclination of the rod to the horizontal in the
position of equilibrium, rounded to the nearest minute if necessary.

A8112′∘

B1749′∘

C848′∘

D7211′∘

Q20:

𝐴𝐵𝐶𝐷 is a parallelogram in which 𝐴𝐵=26cm,
𝐵𝐶=14cm, and
𝑚∠𝐴=30∘.
Forces of magnitudes 60 N, 28 N,
60 N, and 28 N are acting along
𝐵𝐴, 𝐵𝐶,
𝐷𝐶, and 𝐷𝐴
respectively to make a system equivalent to a couple.
Find the magnitude of each of the two forces perpendicular
to 𝐴𝐷 that would need to act at 𝐴 and 𝐷 in order to make an equivalent system.

A30 N,
30 N

B2 N,
2 N

C56 N,
56 N

D4 N,
4 N

Q21:

𝐴𝐵𝐶𝐷 is a rectangle, where 𝐴𝐵=3cm and
𝐵𝐶=4cm. Forces of magnitudes 178, 213, 178, and 213 newtons are acting along
𝐵𝐴, 𝐵𝐶,
𝐷𝐶, and 𝐷𝐴
respectively. Two forces of the same magnitude 60 N are
acting at 𝐴 and 𝐶 in the directions of
𝐵𝐷 and 𝐷𝐵
respectively. Determine the magnitudes of two further forces acting at 𝐵
and 𝐷, perpendicular to 𝐵𝐷, for the system to
be equivalent to a couple of magnitude 171 N⋅cm whose moment is in the same direction of the moment
formed by the two equal forces 60 N and
60 N acting at 𝐴 and 𝐶.

A133 N, 133 N

B106 N, 106 N

C48 N, 48 N

D38 N, 38 N

Q22:

𝐴𝐵𝐶𝐷 is a square of side length 90 cm. Forces of magnitudes of
25, 80,
25, and 80 newtons are
acting along the square’s sides, and two forces, each of magnitude 45√2 newtons, are acting at 𝐴 and 𝐶 as shown in
the figure. Calculate the size of the couple which is equivalent to this system, giving your
answer in N⋅cm

Q23:

A light rod, 𝐴𝐵, has length 46 cm and midpoint 𝑂. Two forces measured in newtons and a moment of magnitude 33 N⋅cm act on the rod, perpendicularly to the vertical plane, as shown in the figure. Determine the moment of the resultant couple acting on the rod.

Q24:

𝐴𝐵𝐶𝐷 is a square of side length 5 cm, where four forces, measured in
newtons, are acting on the sides of the square as shown in the figure. If the system of
forces is equivalent to a couple whose magnitude is 65 N⋅cm,
determine all the possible values of 𝐹.

A𝐹=24N, 𝐹=37N

B𝐹=11N, 𝐹=24N

C𝐹=19N, 𝐹=29N

D𝐹=11N, 𝐹=37N

Q25:

In the given figure, sin𝜃=45 and the algebraic measure of the moment of the
resultant couple equals 90−720√3 N⋅cm. Given that the forces are given in newtons, find the value of 𝐹.